Another polynomial homomorphism

SummaryThe current proposals for applying the so called “fast” O(N logaN) algorithms to multivariate polynomials is that the univariate methods be applied recursively, much in the way more conventional algorithms are used. Since the size of the problems is rather large for which a “fastrd algorithm is more efficient than a classical one, the recursive approach compounds this size completely out of any practical range”.The degree homomorphism is proposed here as an alternative to this recursive approach. It is argued that methods based on the degree homomorphism and a “fast” algorithm may be viable alternatives to more conventional algorithms for certain multivariate problems in the setting of algebraic manipulation. Several such problems are discussed including: polynomial multiplication, powering, division (both exact and with remainder), greatest common divisors and factoring.